Exact criticality condition for randomly layered Ising models with competing interactions on a square lattice

Ising models, with quenched bond disorder along one direction on a square lattice, are studied by the transfer-matrix method. The exact criticality condition is rederived, and extended to an arbitrary variation, in magnitude and/or sign, of the interactions from layer to layer. With competing interactions, a phase diagram is obtained, with a zero-temperature spin-glass point. We consider $2n$ different nearest-neighbor couplings layered and repeated periodically along one direction. The phase boundaries, including zero-temperature bicritical points, the order parameters, and the ground-state degeneracies are illustrated by a detailed discussion for $n=2\mathrm{}$. Closed-form free energies are given for $n=2\mathrm{} \mathrm{and} 3$. The unique criticality condition is derived for any $n$, the limit $n\to \infty$ corresponding to quenched disorder. Our extended result is made possible by a similarity transformation in the free-fermion representation. A conjecture emerges for the criticality condition of randomly layered ferromagnetic $q$-state Potts models.